On Boundary Factors and Traces of Subgroups of Finite Groups

Guo, Wenbin; Skiba, Alexander N.; Tang, Xingzheng

doi:10.1007/s40304-015-0043-4

Cited by 14 publications

(1 citation statement)

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“…If every sylow subgroup is weakly s-permutable in G, then G is soluble. Recall that a normal subgroup E of G is called hypercyclically embedded in G and is denoted by E ≤ Z U (G) (see [18, p. 217]) if either E = 1 or E = 1 and every chief factor of G below E is cyclic, where the symbol Z U (G) is the U-hypercentre of G, that is, the product of all normal hypercyclically embedded subgroups of G. Hypercyclically embedded subgroups play an important role in the theory of groups (see [7,8,18,19] ) and the conditions under which a normal subgroup is hypercyclically embedded in G were found by many authors (see the books [7,8,18,19] and the recent papers [10,14,[20][21][22][23] ).…”

Section: Introductionmentioning

confidence: 99%

On weakly $sigma$-permutable subgroups of finite groups

Zhang¹,

Wu²,

Guo³

2016

Preprint

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Let G be a finite group and σ = {σ i |i ∈ I} be a partition of the set of all primesH for all A ∈ H and all x ∈ G. We say that a subgroup H of G is weakly σ-permutable in G if there exists a σ-subnormal subgroup T of G such that G = HT and H ∩ T ≤ H σG , where H σG is the subgroup of H generated by all those subgroups of H which are σpermutable in G. By using this new notion, we establish some new criterias for a group G to be a σ-soluble and supersoluble, and also we give the conditions under which a normal subgroup of G is hypercyclically embedded.

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Section: Introductionmentioning

confidence: 99%

On weakly $sigma$-permutable subgroups of finite groups

Zhang¹,

Wu²,

Guo³

2016

Preprint

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On an open problem of Guo-Skiba

Guo

2016

Front. Math. China

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Let σ = {σ i |i ∈ I} be some partition of the set P of all primes, that is, P = i∈I σ i and σ i ∩ σ j = ∅ for all i = j. Let G be a finite group. A set H of subgroups of G is said to be a complete Hall σ-set of G if every non-identity member of H is a Hall σ i -subgroup of G and H contains exactly one Hall σ i -subgroup of G for every σ i ∈ σ(G). G is said to be a σ-group if it possesses a complete Hall σ-set. A σ-group G is said to be σ-dispersive provided G has a normal series 1 = G 1 < G 2 < · · · < G t < G t+1 = G and a complete Hall σ-set {H 1 , H 2 , · · · , H t } such that G i H i = G i+1 for all i = 1, 2, . . . t. In this paper, we give a characterizations of σ-dispersive group, which give a positive answer to an open problem of Skiba in the paper [1].

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